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arXiv:2110.11327 · The Journal of Chemical Physics

Efficient fully-coherent quantum signal processing algorithms for real-time dynamics simulation

John M. Martyn, Yuan Liu, Zachary E. Chin, Isaac L. Chuang

Published Jan 10, 2023• 54 citations
arXivHTMLPDF

Abstract

Simulating the unitary dynamics of a quantum system is a fundamental problem of quantum mechanics, in which quantum computers are believed to have significant advantage over their classical counterparts. One prominent such instance is the simulation of electronic dynamics, which plays an essential role in chemical reactions, non-equilibrium dynamics, and material design. These systems are time-dependent, which requires that the corresponding simulation algorithm can be successfully concatenated with itself over different time intervals to reproduce the overall coherent quantum dynamics of the system. In this paper, we quantify such simulation algorithms by the property of being fully-coherent: the algorithm succeeds with arbitrarily high success probability 1 − δ while only requiring a single copy of the initial state. We subsequently develop fully-coherent simulation algorithms based on quantum signal processing (QSP), including a novel algorithm that circumvents the use of amplitude amplification while also achieving a query complexity additive in time t, ln(1/δ), and ln(1/ϵ) for error tolerance ϵ: Θ‖H‖|t|+ln(1/ϵ)+ln(1/δ). Furthermore, we numerically analyze these algorithms by applying them to the simulation of the spin dynamics of the Heisenberg model and the correlated electronic dynamics of an H2 molecule. Since any electronic Hamiltonian can be mapped to a spin Hamiltonian, our algorithm can efficiently simulate time-dependent ab initio electronic dynamics in the circuit model of quantum computation. Accordingly, it is also our hope that the present work serves as a bridge between QSP-based quantum algorithms and chemical dynamics, stimulating a cross-fertilization between these exciting fields.

DOI
10.1063/5.0124385

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Simulating the unitary dynamics of a quantum system is a fundamental problem of quantum mechanics, in which quantum computers are believed to have significant advantage over their classical counterparts. One prominent such instance is the simulation of electronic dynamics, which plays an essential role in chemical reactions, non-equilibrium dynamics, and material design. These systems are time-dependent, which requires that the corresponding simulation algorithm can be successfully concatenated with itself over different time intervals to reproduce the overall coherent quantum dynamics of the system. In this paper, we quantify such simulation algorithms by the property of being fully-coherent: the algorithm succeeds with arbitrarily high success probability 1 − δ while only requiring a single copy of the initial state. We subsequently develop fully-coherent simulation algorithms based on quantum signal processing (QSP), including a novel algorithm that circumvents the use of amplitude amplification while also achieving a query complexity additive in time t, ln(1/δ), and ln(1/ϵ) for error tolerance ϵ: Θ‖H‖|t|+ln(1/ϵ)+ln(1/δ). Furthermore, we numerically analyze these algorithms by applying them to the simulation of the spin dynamics of the Heisenberg model and the correlated electronic dynamics of an H2 molecule. Since any electronic Hamiltonian can be mapped to a spin Hamiltonian, our algorithm can efficiently simulate time-dependent ab initio electronic dynamics in the circuit model of quantum computation. Accordingly, it is also our hope that the present work serves as a bridge between QSP-based quantum algorithms and chemical dynamics, stimulating a cross-fertilization between these exciting fields.
Hamiltonian simulation is a fundamental problem at the heart of quantum computation, and the associated simulation algorithms are useful building blocks for designing larger quantum algorithms. In order to be successfully concatenated into a larger quantum algorithm, a Hamiltonian simulation algorithm must succeed with arbitrarily high success probability 1−δ1-\delta1−δ while only requiring a single copy of the initial state, a property which we call fully-coherent. Although optimal Hamiltonian simulation has been achieved by quantum signal processing (QSP), with query complexity linear in time ttt and logarithmic in inverse error ln⁡(1/ϵ)\ln(1/\epsilon)ln(1/ϵ), the corresponding algorithm is not fully-coherent as it only succeeds with probability close to 1/41/41/4. While this simulation algorithm can be made fully-coherent by employing amplitude amplification at the expense of appending a ln⁡(1/δ)\ln(1/\delta)ln(1/δ) multiplicative factor to the query complexity, here we develop a new fully-coherent Hamiltonian simulation algorithm that achieves a query complexity additive in ln⁡(1/δ)\ln(1/\delta)ln(1/δ): Θ(∥H∥∣t∣+ln⁡(1/ϵ)+ln⁡(1/δ))\Theta\big( \|\mathcal{H}\| |t| + \ln(1/\epsilon) + \ln(1/\delta)\big)Θ(∥H∥∣t∣+ln(1/ϵ)+ln(1/δ)). We accomplish this by compressing the spectrum of the Hamiltonian with an affine transformation, and applying to it a QSP polynomial that approximates the complex exponential only over the range of the compressed spectrum. We further numerically analyze the complexity of this algorithm and demonstrate its application to the simulation of the Heisenberg model in constant and time-dependent external magnetic fields. We believe that this efficient fully-coherent Hamiltonian simulation algorithm can serve as a useful subroutine in quantum algorithms where maintaining coherence is paramount.
The physics of quantum mechanics is the inspiration for, and underlies, quantum computation. As such, one expects physical intuition to be highly influential in the understanding and design of many quantum algorithms, particularly simulation of physical systems. Surprisingly, this has been challenging, with current Hamiltonian simulation algorithms remaining abstract and often the result of sophisticated but unintuitive constructions. We contend that physical intuition can lead to optimal simulation methods by showing that a focus on simple single-qubit rotations elegantly furnishes an optimal algorithm for Hamiltonian simulation, a universal problem that encapsulates all the power of quantum computation. Specifically, we show that the query complexity of implementing time evolution by a d-sparse Hamiltonian H[over ^] for time-interval t with error ε is O[td∥H[over ^]∥_{max}+log(1/ε)/loglog(1/ε)], which matches lower bounds in all parameters. This connection is made through general three-step "quantum signal processing" methodology, comprised of (i) transducing eigenvalues of H[over ^] into a single ancilla qubit, (ii) transforming these eigenvalues through an optimal-length sequence of single-qubit rotations, and (iii) projecting this ancilla with near unity success probability.
Simulations of chemical dynamics are a powerful means for understanding chemistry. However, classical computers struggle to simulate many chemical processes, especially non-adiabatic ones, where the Born-Oppenheimer approximation breaks down. Quantum computers could simulate quantum-chemical dynamics more efficiently than classical computers, but there is currently no complete quantum algorithm for calculating dynamical observables to within a known error. Here, we develop an efficient, end-to-end quantum algorithm for simulating chemical dynamics that avoids all uncontrolled approximations (including the Born-Oppenheimer approximation) and whose error is bounded subject to mild assumptions. To do so, we treat the nuclei and the electrons on an equal footing and simulate the full molecular wavefunction on a momentum-space grid in first quantization, including all algorithmic steps: initial-state preparation, time evolution using qubitization, and measurement of chemical observables such as reaction yields and rates. Our work gives the first algorithm for quantum simulation of chemistry whose end-to-end complexity achieves sublinear scaling in the size of the grid. We achieve this by developing an exponentially faster method for initial-state-preparation. Photochemistry is a likely early application of our algorithm and we estimate resources required for end-to-end simulations of non-adiabatic dynamics of atmospherically important molecules. Classically intractable photochemical computations could be performed using resources comparable to those required for other chemical applications of quantum computing.
Quantum computers are expected to significantly outperform classical computers for certain computational tasks. One of the most promising applications is the simulation of many-body systems such as complex molecules or materials. In this thesis, we present several quantum algorithms for simulating many-body systems that can provide considerably better scaling than prior approaches. First, we describe a fully coherent quantum algorithm for simulating molecular dynamics within the Born-Oppenheimer approximation that leads to super-polynomially better scaling with respect to the error tolerance than previous work. Our algorithm utilizes the Koopman-von Neumann formulation of classical mechanics. This allows us to express the classical equations of motion as unitary dynamics which can be efficiently simulated on a quantum computer. We demonstrate that these simulations can be performed either at constant energy or at constant temperature. The latter allows us to estimate thermodynamic properties such as the Helmholtz free energy of the system. Next, we present two closely related quantum algorithms for solving continuous optimization problems such as finding the equilibrium configurations of molecules. These algorithms do not require any explicit gradient estimations. Instead, we encode the optimization problem into the dynamics of a quantum or classical system with friction. The dynamics are governed by a carefully chosen time-dependent Hamiltonian and thus can be simulated using efficient Hamiltonian simulation algorithms. We provide evidence that our fully coherent quantum algorithms can, in certain cases, provide exponentially better asymptotic scaling than approaches based on gradient descent. Lastly, we present a quantum algorithm for preparing the ground state of a set of interacting systems that can circumvent the exponential scaling suggested by the Van Vleck catastrophe. Our algorithm utilizes a divide-and-conquer strategy that repeatedly uses phase estimation to merge ground states of increasingly larger subsystems. We provide rigorous bounds for this approach and show that, under certain assumptions, our algorithm scales only quasi-polynomially with the number of systems.
We present a comprehensive end-to-end framework for simulating the real-time dynamics of chemical systems on a fault-tolerant quantum computer, incorporating both electronic and nuclear quantum degrees of freedom. An all-particle simulation is nominally efficient on a quantum computer, but practically infeasible. Hence, central to our approach is the construction of a first-quantized plane-wave algorithm making use of pseudoions. The latter consolidate chemically inactive electrons and the nucleus into a single effective dynamical ionic entity, extending the well-established concept of pseudopotentials in quantum chemistry to a two-body interaction. We explicitly describe efficient quantum circuits for initial state preparation across all degrees of freedom, as well as for block-encoding the Hamiltonian describing interacting pseudoions and chemically active electrons, by leveraging recent advances in quantum rejection sampling to optimize the implementations. To extract useful chemical information, we first design molecular fingerprints by combining density-functional calculations with machine learning techniques, and subsequently validate them through surrogate classical molecular dynamics simulations. These fingerprints are then coherently encoded on a quantum computer for efficient molecular identification via amplitude estimation. We provide an extensive analysis of the cost of running the algorithm on a fault-tolerant quantum computer for several chemically interesting systems. As an illustration, simulating the interaction between NH3\mathrm{NH_3}NH3​ and BF3\mathrm{BF_3}BF3​ (a 40-particle system) requires 808 logical qubits to encode the problem, and approximately 101110^{11}1011 Toffoli gates per femtosecond of time evolution. Our results establish a foundation for further quantum algorithm development targeting chemical and material dynamics.
Simulations of quantum chemistry and quantum materials are believed to be among the most important applications of quantum information processors. However, realizing practical quantum advantage for such problems is challenging because of the prohibitive computational cost of programming typical problems into quantum hardware. Here we introduce a simulation framework for strongly correlated quantum systems represented by model spin Hamiltonians that uses reconfigurable qubit architectures to simulate real-time dynamics in a programmable way. Our approach also introduces an algorithm for extracting chemically relevant spectral properties via classical co-processing of quantum measurement results. We develop a digital–analogue simulation toolbox for efficient Hamiltonian time evolution using digital Floquet engineering and hardware-optimized multi-qubit operations to accurately realize complex spin–spin interactions. As an example, we propose an implementation based on Rydberg atom arrays. In addition, we show how detailed spectral information can be extracted from the dynamics through snapshot measurements and single-ancilla control, enabling the evaluation of excitation energies and finite-temperature susceptibilities from a single dataset. To illustrate the approach, we show how to use the method to compute key properties of a polynuclear transition-metal catalyst and two-dimensional magnetic materials. Quantum simulations of chemistry and materials are challenging due to the complexity of correlated systems. A framework based on reconfigurable qubit architectures and digital–analogue simulations provides a hardware-efficient path forwards.